Theorem iota0def 43347

Description: Example for a defined iota being the empty set, i.e., ∀𝑦𝑥 ⊆ 𝑦 is a wff satisfied by a unique value 𝑥, namely 𝑥 = ∅ (the empty set is the one and only set which is a subset of every set). (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
iota0def	⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Distinct variable group:   𝑥,𝑦

Proof of Theorem iota0def

Step	Hyp	Ref	Expression
1		0ex 5204	. 2 ⊢ ∅ ∈ V
2		al0ssb 5205	. . . 4 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)
3	2	a1i 11	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅))
4	3	iota5 6331	. 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅)
5	1, 1, 4	mp2an 690	1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅

Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1534   = wceq 1536   ∈ wcel 2113  Vcvv 3491   ⊆ wss 3929  ∅c0 4284  ℩cio 6305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-nul 5203

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-sn 4561  df-pr 4563  df-uni 4832  df-iota 6307

This theorem is referenced by: (None)